Subgroups — #45655206

Order 32576.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22550861	C₂	1	1	yes	maximal, characteristic
2	C₂²×#16861325	C₂	1	1	yes	maximal, characteristic, complemented
3	#30654127	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550861	—	2	2	no	—
6	#22550861	—	2	2	no	—
7	C₂×Dic₁₀₁₈	C₄	1	1	yes	characteristic
8	C₂×D₂₀₃₆	C₄	1	1	yes	characteristic
9	C₂×C₂²⋊D₅₀₉	C₄	1	2	yes	—
10	C₂×C₄×D₅₀₉	C₂²	1	2	yes	—
11	C₂×#16861325	—	2	2	no	—
12	C₂×#16861325	C₂²	1	2	yes	—
13	C₂²×C₄×C₅₀₉	C₂²	1	1	yes	characteristic
14	M₁₆×C₅₀₉	—	2	2	no	—
15	C₂×C₈×C₅₀₉	C₂²	1	2	yes	—
16	C₂²×D₅₀₉	C₂×C₄	1	2	yes	—
17	Dic₁₀₁₈	—	2	2	no	—
18	D₂₀₃₆	—	2	2	no	—
19	C₂²⋊D₅₀₉	—	2	4	no	—
20	C₂×Dic₅₀₉	C₂×C₄	1	2	yes	—
21	C₄×D₅₀₉	—	2	2	no	—
22	C₄×D₅₀₉	—	2	2	no	—
23	#16861325	—	2	2	no	—
24	C₂³×C₅₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₀₉	D₄	1	2	yes	—
27	C₂×C₄×C₅₀₉	D₄	1	2	yes	—
28	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
29	C₈×C₅₀₉	—	2	2	no	—
30	C₂²×C₅₀₉	—	2	2	no	—
31	C₂²×C₅₀₉	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×D₅₀₉	—	2	2	no	—
34	C₂×D₅₀₉	—	2	4	no	—
35	C₄×C₅₀₉	C₂²⋊C₄	1	2	yes	—
36	C₄×C₅₀₉	C₂×D₄	1	1	yes	characteristic
37	C₄×C₅₀₉	C₂×D₄	1	1	yes	characteristic
38	Dic₅₀₉	—	2	2	no	—
39	C₂×C₅₀₉	—	2	2	no	—
40	C₂×C₅₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₅₀₉	#131	1	2	yes	—
42	D₅₀₉	—	4	2	no	—
43	C₅₀₉	#408	1	1	yes	characteristic, complemented, Sylow
44	#408	—	509	1	no	maximal, Sylow
45	C₂×#41	—	509	1	no	—
46	C₂×M₁₆	D₅₀₉	1	1	yes	characteristic, complemented
47	C₂²×C₈	—	509	1	no	—
48	C₂²⋊C₈	—	509	4	no	—
49	#41	—	1018	2	no	—
50	#41	—	1018	2	no	—
51	C₂×Q₈	—	509	1	no	—
52	C₂×D₄	—	509	2	no	—
53	C₂×D₄	—	509	1	no	—
54	C₂²×C₄	—	509	2	no	—
55	C₂²×C₄	C₂×D₅₀₉	1	1	yes	characteristic
56	M₁₆	—	2	2	no	—
57	C₂×C₈	—	1018	2	no	—
58	C₂×C₈	—	509	2	no	—
59	C₂×C₈	C₂×D₅₀₉	1	2	yes	—
60	C₂³	—	509	2	no	—
61	C₂³	C₄×D₅₀₉	1	1	yes	characteristic
62	Q₈	—	1018	2	no	—
63	D₄	—	1018	4	no	—
64	D₄	—	1018	2	no	—
65	C₂×C₄	—	1018	2	no	—
66	C₂×C₄	—	1018	2	no	—
67	C₂×C₄	—	509	2	no	—
68	C₂×C₄	C₄×D₅₀₉	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₅₀₉	1	2	yes	—
70	C₂×C₄	D₂₀₃₆	1	2	yes	—
71	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	1018	2	no	—
74	C₂²	—	2	2	no	—
75	C₂²	—	1018	4	no	—
76	C₂²	—	1018	2	no	—
77	C₂²	#22550838	1	2	yes	—
78	C₂²	C₂×C₄×D₅₀₉	1	1	yes	characteristic
79	C₄	—	1018	2	no	—
80	C₄	#22550838	1	2	yes	—
81	C₄	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
82	C₄	C₂×D₂₀₃₆	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	2036	2	no	—
85	C₂	C₂×#22550838	1	1	yes	characteristic
86	C₂	#30654223	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₀₉
commutator, socle	C₂²×C₅₀₉	C₂²×C₄
Fitting	C₂×M₁₆×C₅₀₉	C₂
prime radical	C₂×#22550861	C₂
cosocle	C₂×C₄×C₅₀₉	C₂³
hypercenter	C₂×M₁₆	D₅₀₉
hypocenter	C₅₀₉	#408
LCS	G ▷ C₂²×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×#16861325	C₂	semidirect
3. #30654127	C₂	semidirect
4. C₂×M₁₆×C₅₀₉	C₂	semidirect
43. C₅₀₉	#408	semidirect
46. C₂×M₁₆	D₅₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#22550861	2
C₂²×#16861325	2
#30654127	2
C₂×M₁₆×C₅₀₉	2
#408	509